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But the word Geometry emphasizes the fact that our 

 experience does include a knowledge of things which 

 are sometimes very nearly indeed like Euclid's triangles. 

 Such experimental triangles play a great part in many 

 scientific investigations, and I cannot find a better way of 

 showing you the difference between attainable experi- 

 mental exactness and the absolute exactness which results 

 from following accepted definitions to their logical con- 

 clusion, than by saying a word or two about our know- 

 ledge of these experimental triangles. If we measure the 

 three angles of one such triangle as accurately as we can, 

 and add the results together, we shall find that the sum 

 of these angles is very nearly indeed equal to two right- 

 angles ; but it will generally differ by some small amount 

 from the exact sum which we find when we deal mentally 

 with Euclid's triangles. The first case of such a measure- 

 ment I found, when I looked for an example of this, was 

 one in which the observed sum of the three angles con- 

 tained 1 80 o' 12", instead of exactly 180. This is a 

 very close approximation to Euclid's result, differing 

 from it by only one part in 54,000, but it is not exactly 

 Euclid's result, and the question arises, whether this small 

 difference between one result and the other depends on 

 some difference between the properties of the two kinds 

 of triangle or not. Now the natural thing for an observer 

 to do, after he had found that his sum differed from 

 Euclid's by twelve seconds of arc, would be to measure his 

 triangle again, in order to see if he had made any mistake. 

 If he did this, it is very unlikely that he would find 

 exactly 180, but it is also very unlikely that he would find 

 exactly 180 o' 12". If he made a great number of mea- 

 surements, he would find that some one or more of them 

 gave every possible value within a small range, outside 

 which none of them would fall ; and within this small 



